BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Tim Stokes (University of Waikato) DTSTART:20200513T103000Z DTEND:20200513T113000Z DTSTAMP:20260423T010944Z UID:YSseminar/2 DESCRIPTION:Title: How to generalise demonic composition\nby Tim Stokes (University of Waikato) as part of York semigroup seminar\n\n\nAbstract\nDemonic composit ion is defined on the set of binary relations over the non-empty set $X$\, $Rel_X$\, and is a variant of standard or ``angelic" composition. It ari ses naturally in the setting of the theory of non-deterministic computer p rograms\, and shares many of the nice features of ordinary composition (it is associative\, and generalises composition of functions). When equippe d with the operations of demonic composition and domain\, the resulting un ary semigroup defined on $Rel_X$ is a left restriction semigroup (like $PT _X$\, the semigroup of partial functions on $X$)\, whereas usual compositi on and domain give a unary semigroup satisfying weaker laws. \n\n\nBy con structing a constellation (a kind of ``one-sided" category)\, we show how this secondary demonic left restriction semigroup structure arises on $Rel _X$\, placing it in a more general context. The construction applies to a ny unary semigroup with a ``domain-like" operation satisfying certain mini mal conditions which we identify. \n\nIn particular it is shown that any Baer $*$-semigroup $S$ can be given a left restriction semigroup structure using the construction\, and that the result is even an inverse semigroup if $S$ is $*$-regular. It follows that the semigroup of $n\\times n$ mat rices over the real or complex numbers is an inverse semigroup with respec t to a modified notion of product that almost always agrees with the usual matrix product\, and in which inverse is pseudoinverse (Moore-Penrose inv erse).\n LOCATION:https://researchseminars.org/talk/YSseminar/2/ END:VEVENT END:VCALENDAR